# Properties

 Label 63.4.e.a Level $63$ Weight $4$ Character orbit 63.e Analytic conductor $3.717$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [63,4,Mod(37,63)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(63, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("63.37");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 63.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$3.71712033036$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (3 \zeta_{6} - 3) q^{2} - \zeta_{6} q^{4} + (3 \zeta_{6} - 3) q^{5} + (21 \zeta_{6} - 14) q^{7} - 21 q^{8} +O(q^{10})$$ q + (3*z - 3) * q^2 - z * q^4 + (3*z - 3) * q^5 + (21*z - 14) * q^7 - 21 * q^8 $$q + (3 \zeta_{6} - 3) q^{2} - \zeta_{6} q^{4} + (3 \zeta_{6} - 3) q^{5} + (21 \zeta_{6} - 14) q^{7} - 21 q^{8} - 9 \zeta_{6} q^{10} - 15 \zeta_{6} q^{11} - 64 q^{13} + ( - 42 \zeta_{6} - 21) q^{14} + ( - 71 \zeta_{6} + 71) q^{16} + 84 \zeta_{6} q^{17} + ( - 16 \zeta_{6} + 16) q^{19} + 3 q^{20} + 45 q^{22} + (84 \zeta_{6} - 84) q^{23} + 116 \zeta_{6} q^{25} + ( - 192 \zeta_{6} + 192) q^{26} + ( - 7 \zeta_{6} + 21) q^{28} + 297 q^{29} + 253 \zeta_{6} q^{31} + 45 \zeta_{6} q^{32} - 252 q^{34} + ( - 42 \zeta_{6} - 21) q^{35} + ( - 316 \zeta_{6} + 316) q^{37} + 48 \zeta_{6} q^{38} + ( - 63 \zeta_{6} + 63) q^{40} - 360 q^{41} + 26 q^{43} + (15 \zeta_{6} - 15) q^{44} - 252 \zeta_{6} q^{46} + (30 \zeta_{6} - 30) q^{47} + ( - 147 \zeta_{6} - 245) q^{49} - 348 q^{50} + 64 \zeta_{6} q^{52} + 363 \zeta_{6} q^{53} + 45 q^{55} + ( - 441 \zeta_{6} + 294) q^{56} + (891 \zeta_{6} - 891) q^{58} - 15 \zeta_{6} q^{59} + ( - 118 \zeta_{6} + 118) q^{61} - 759 q^{62} + 433 q^{64} + ( - 192 \zeta_{6} + 192) q^{65} + 370 \zeta_{6} q^{67} + ( - 84 \zeta_{6} + 84) q^{68} + ( - 63 \zeta_{6} + 189) q^{70} + 342 q^{71} - 362 \zeta_{6} q^{73} + 948 \zeta_{6} q^{74} - 16 q^{76} + ( - 105 \zeta_{6} + 315) q^{77} + (467 \zeta_{6} - 467) q^{79} + 213 \zeta_{6} q^{80} + ( - 1080 \zeta_{6} + 1080) q^{82} - 477 q^{83} - 252 q^{85} + (78 \zeta_{6} - 78) q^{86} + 315 \zeta_{6} q^{88} + ( - 906 \zeta_{6} + 906) q^{89} + ( - 1344 \zeta_{6} + 896) q^{91} + 84 q^{92} - 90 \zeta_{6} q^{94} + 48 \zeta_{6} q^{95} + 503 q^{97} + ( - 735 \zeta_{6} + 1176) q^{98} +O(q^{100})$$ q + (3*z - 3) * q^2 - z * q^4 + (3*z - 3) * q^5 + (21*z - 14) * q^7 - 21 * q^8 - 9*z * q^10 - 15*z * q^11 - 64 * q^13 + (-42*z - 21) * q^14 + (-71*z + 71) * q^16 + 84*z * q^17 + (-16*z + 16) * q^19 + 3 * q^20 + 45 * q^22 + (84*z - 84) * q^23 + 116*z * q^25 + (-192*z + 192) * q^26 + (-7*z + 21) * q^28 + 297 * q^29 + 253*z * q^31 + 45*z * q^32 - 252 * q^34 + (-42*z - 21) * q^35 + (-316*z + 316) * q^37 + 48*z * q^38 + (-63*z + 63) * q^40 - 360 * q^41 + 26 * q^43 + (15*z - 15) * q^44 - 252*z * q^46 + (30*z - 30) * q^47 + (-147*z - 245) * q^49 - 348 * q^50 + 64*z * q^52 + 363*z * q^53 + 45 * q^55 + (-441*z + 294) * q^56 + (891*z - 891) * q^58 - 15*z * q^59 + (-118*z + 118) * q^61 - 759 * q^62 + 433 * q^64 + (-192*z + 192) * q^65 + 370*z * q^67 + (-84*z + 84) * q^68 + (-63*z + 189) * q^70 + 342 * q^71 - 362*z * q^73 + 948*z * q^74 - 16 * q^76 + (-105*z + 315) * q^77 + (467*z - 467) * q^79 + 213*z * q^80 + (-1080*z + 1080) * q^82 - 477 * q^83 - 252 * q^85 + (78*z - 78) * q^86 + 315*z * q^88 + (-906*z + 906) * q^89 + (-1344*z + 896) * q^91 + 84 * q^92 - 90*z * q^94 + 48*z * q^95 + 503 * q^97 + (-735*z + 1176) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} - q^{4} - 3 q^{5} - 7 q^{7} - 42 q^{8}+O(q^{10})$$ 2 * q - 3 * q^2 - q^4 - 3 * q^5 - 7 * q^7 - 42 * q^8 $$2 q - 3 q^{2} - q^{4} - 3 q^{5} - 7 q^{7} - 42 q^{8} - 9 q^{10} - 15 q^{11} - 128 q^{13} - 84 q^{14} + 71 q^{16} + 84 q^{17} + 16 q^{19} + 6 q^{20} + 90 q^{22} - 84 q^{23} + 116 q^{25} + 192 q^{26} + 35 q^{28} + 594 q^{29} + 253 q^{31} + 45 q^{32} - 504 q^{34} - 84 q^{35} + 316 q^{37} + 48 q^{38} + 63 q^{40} - 720 q^{41} + 52 q^{43} - 15 q^{44} - 252 q^{46} - 30 q^{47} - 637 q^{49} - 696 q^{50} + 64 q^{52} + 363 q^{53} + 90 q^{55} + 147 q^{56} - 891 q^{58} - 15 q^{59} + 118 q^{61} - 1518 q^{62} + 866 q^{64} + 192 q^{65} + 370 q^{67} + 84 q^{68} + 315 q^{70} + 684 q^{71} - 362 q^{73} + 948 q^{74} - 32 q^{76} + 525 q^{77} - 467 q^{79} + 213 q^{80} + 1080 q^{82} - 954 q^{83} - 504 q^{85} - 78 q^{86} + 315 q^{88} + 906 q^{89} + 448 q^{91} + 168 q^{92} - 90 q^{94} + 48 q^{95} + 1006 q^{97} + 1617 q^{98}+O(q^{100})$$ 2 * q - 3 * q^2 - q^4 - 3 * q^5 - 7 * q^7 - 42 * q^8 - 9 * q^10 - 15 * q^11 - 128 * q^13 - 84 * q^14 + 71 * q^16 + 84 * q^17 + 16 * q^19 + 6 * q^20 + 90 * q^22 - 84 * q^23 + 116 * q^25 + 192 * q^26 + 35 * q^28 + 594 * q^29 + 253 * q^31 + 45 * q^32 - 504 * q^34 - 84 * q^35 + 316 * q^37 + 48 * q^38 + 63 * q^40 - 720 * q^41 + 52 * q^43 - 15 * q^44 - 252 * q^46 - 30 * q^47 - 637 * q^49 - 696 * q^50 + 64 * q^52 + 363 * q^53 + 90 * q^55 + 147 * q^56 - 891 * q^58 - 15 * q^59 + 118 * q^61 - 1518 * q^62 + 866 * q^64 + 192 * q^65 + 370 * q^67 + 84 * q^68 + 315 * q^70 + 684 * q^71 - 362 * q^73 + 948 * q^74 - 32 * q^76 + 525 * q^77 - 467 * q^79 + 213 * q^80 + 1080 * q^82 - 954 * q^83 - 504 * q^85 - 78 * q^86 + 315 * q^88 + 906 * q^89 + 448 * q^91 + 168 * q^92 - 90 * q^94 + 48 * q^95 + 1006 * q^97 + 1617 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/63\mathbb{Z}\right)^\times$$.

 $$n$$ $$10$$ $$29$$ $$\chi(n)$$ $$-1 + \zeta_{6}$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.50000 + 2.59808i 0 −0.500000 0.866025i −1.50000 + 2.59808i 0 −3.50000 + 18.1865i −21.0000 0 −4.50000 7.79423i
46.1 −1.50000 2.59808i 0 −0.500000 + 0.866025i −1.50000 2.59808i 0 −3.50000 18.1865i −21.0000 0 −4.50000 + 7.79423i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.4.e.a 2
3.b odd 2 1 21.4.e.a 2
7.b odd 2 1 441.4.e.c 2
7.c even 3 1 inner 63.4.e.a 2
7.c even 3 1 441.4.a.l 1
7.d odd 6 1 441.4.a.k 1
7.d odd 6 1 441.4.e.c 2
12.b even 2 1 336.4.q.e 2
21.c even 2 1 147.4.e.h 2
21.g even 6 1 147.4.a.a 1
21.g even 6 1 147.4.e.h 2
21.h odd 6 1 21.4.e.a 2
21.h odd 6 1 147.4.a.b 1
84.j odd 6 1 2352.4.a.bd 1
84.n even 6 1 336.4.q.e 2
84.n even 6 1 2352.4.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.e.a 2 3.b odd 2 1
21.4.e.a 2 21.h odd 6 1
63.4.e.a 2 1.a even 1 1 trivial
63.4.e.a 2 7.c even 3 1 inner
147.4.a.a 1 21.g even 6 1
147.4.a.b 1 21.h odd 6 1
147.4.e.h 2 21.c even 2 1
147.4.e.h 2 21.g even 6 1
336.4.q.e 2 12.b even 2 1
336.4.q.e 2 84.n even 6 1
441.4.a.k 1 7.d odd 6 1
441.4.a.l 1 7.c even 3 1
441.4.e.c 2 7.b odd 2 1
441.4.e.c 2 7.d odd 6 1
2352.4.a.i 1 84.n even 6 1
2352.4.a.bd 1 84.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 3T_{2} + 9$$ acting on $$S_{4}^{\mathrm{new}}(63, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 3T + 9$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 3T + 9$$
$7$ $$T^{2} + 7T + 343$$
$11$ $$T^{2} + 15T + 225$$
$13$ $$(T + 64)^{2}$$
$17$ $$T^{2} - 84T + 7056$$
$19$ $$T^{2} - 16T + 256$$
$23$ $$T^{2} + 84T + 7056$$
$29$ $$(T - 297)^{2}$$
$31$ $$T^{2} - 253T + 64009$$
$37$ $$T^{2} - 316T + 99856$$
$41$ $$(T + 360)^{2}$$
$43$ $$(T - 26)^{2}$$
$47$ $$T^{2} + 30T + 900$$
$53$ $$T^{2} - 363T + 131769$$
$59$ $$T^{2} + 15T + 225$$
$61$ $$T^{2} - 118T + 13924$$
$67$ $$T^{2} - 370T + 136900$$
$71$ $$(T - 342)^{2}$$
$73$ $$T^{2} + 362T + 131044$$
$79$ $$T^{2} + 467T + 218089$$
$83$ $$(T + 477)^{2}$$
$89$ $$T^{2} - 906T + 820836$$
$97$ $$(T - 503)^{2}$$